Thermodynamic Origin of the Photostability of the Two-Dimensional Perovskite PEA2Pb(I1–xBrx)4

The two-dimensional (2D) mixed halide perovskite PEA2Pb(I1–xBrx)4 exhibits high phase stability under illumination as compared to the three-dimensional (3D) counterpart MAPb(I1–xBrx)3. We explain this difference using a thermodynamic theory that considers the sum of a compositional and a photocarrier free energy. Ab initio calculations show that the improved compositional phase stability of the 2D perovskite is caused by a preferred I–Br distribution, leading to a much lower critical temperature for halide segregation in the dark than for the 3D perovskite. Moreover, a smaller increase of the band gap with Br concentration x and a markedly shorter photocarrier lifetime in the 2D perovskite reduce the driving force for phase segregation under illumination, enhancing the photostability.

M etal-halide perovskites are rapidly emerging as a new class of semiconducting materials for photovoltaics. They have the general chemical formula ABX 3 , where A is a monovalent organic or inorganic cation like methylammonium (MA), formamidinium (FA), or Cs, M is a divalent metal cation like Pb or Sn, and X is a halide anion like I, Br, or Cl. 1−6 By sandwiching three-dimensional (3D) ABX 3 perovskite layers with large organic spacer cations, layered twodimensional (2D) perovskites can be obtained. 7−11 Among these, Ruddlesden−Popper (RP) perovskites have been widely studied as light absorbers owing to their superior moisture resistance. 12−16 The general chemical formula of RP-type halide perovskites is R 2 A n−1 B n X 3n+1 , where R is a monovalent organic spacer cation, e.g., phenethylammonium (PEA) or butylammonium (BA), 9,10 and n is the number of B−X octahedral layers. Recently, the band gap tunability of 2D RP halide perovskites by compositional alloying on X sites has attracted increasing attention, 17−21 which stimulated the use of these perovskites in solar cells, light-emitting devices, and photodetectors. 8,22−24 Very promising is the recent finding that light-induced halide segregation, which occurs in the 3D perovksite MAPb(I 1−x Br x ) 3 perovskite when the Br concentration x exceeds about 0.2, 25,26 is absent in the 2D n = 1 perovskite PEA 2 Pb(I 0.5 Br 0.5 ) 4 . 17,18 A kinetic explanation has been given of this improved photostability based on an increased halide migration barrier in PEA 2 Pb(I 0.5 Br 0.5 ) 4 in comparison to MAPb(I 0.5 Br 0.5 ) 3 , by about 80 meV. 17 Although such an increased energy barrier�about 3 times the roomtemperature thermal energy of 25 meV�will definitely slow down halide segregation, it cannot explain the complete absence of such segregation. Instead, in this Letter, we will provide an equilibrium thermodynamic explanation for the increased photostability of PEA 2 Pb(I 0.5 Br 0.5 ) 4 . To understand light-induced halide segregation in 3D mixed halide perovskites, we recently developed a unified thermodynamic theory that considers a free energy that is the sum of a compositional and a photocarrier free energy. 27 In this theory, photocarriers can decrease their free energy by funneling into a low-band-gap phase that is nucleated out of a mixed parent phase. We applied the theory to a series of 3D mixed I−Br perovskites and could explain several experimental observations from the calculated composition−temperature (x−T) phase diagrams at different illumination intensities. 27 For MAPb(I 1−x Br x ) 3 , we predicted a dependence of the threshold illumination intensity on composition and temperature and a temperature dependence of the threshold Br concentration for halide demixing that are qualitatively consistent with experimental results. 27 In this Letter, we apply this thermodynamic theory to the 2D mixed halide perovskite PEA 2 Pb(I 1−x Br x ) 4 and study its phase stability in the dark and under illumination. We start with a calculation of the Helmholtz compositional free energy and then construct the x−T phase diagrams of PEA 2 Pb(I 1−x Br x ) 4 in the dark. After adding a photocarrier contribution, we construct the phase diagrams for different illumination intensities. Similar to MAPb(I 1−x Br x ) 3 , we predict the existence of an illumination intensity and Br concentration threshold for halide demixing. We find that, both in the dark and under illumination, PEA 2 Pb(I 1−x Br x ) 4 is thermodynamically much more stable than MAPb(I 1−x Br x ) 3 , and we discuss the reasons for this enhanced stability.
To obtain the Helmholtz compositional mixing free energy of PEA 2 Pb(I 1−x Br x ) 4 , we first calculate within density functional theory (DFT) the mixing enthalpies ΔU per formula unit (f.u.) of all possible I−Br configurations at different Br concentrations x = 0, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, and 1, according to eq 1 in the "Methods" section. The results are shown by circles in Figure 1a. We clearly identify an enthalpically preferred I−Br distribution where the Br anions are located at the equatorial sites of the Pb−X octahedral layers and the I anions at the axial sites. This leads to a situation where the most stable (unstable) configuration with the lowest (highest) enthalpy for each Br concentration x has the maximum number of Br (I) anions at equatorial sites. The configurations with the highest and lowest enthalpy for x = 0.5 are displayed in Figure 1b, where the top configuration is the most unstable (enthalpy given by the pink filled circle in Figure  1a), with all Br anions in the axial layer, and the bottom configuration is the most stable (enthalpy given by the green filled circle), with all Br anions in the equatorial layer.
The preferential I−Br distribution can be explained by a volume effect. Due to the inequivalence of equatorial and axial sites, the substitution of smaller but more electronegative Br anions for I anions at equatorial and axial sites of the unit cell of PEA 2 PbI 4 to form shorter Pb−Br bonds gives rise to different degrees of volume contraction (see Section S1 in the Supporting Information). For a given Br concentration, the volume of the unit cell tends to be smaller when the Br anions are placed at equatorial sites than when they are placed at axial sites. This is mainly due to the fact that each equatorial anion forms chemical bonds with two adjacent Pb cations, whereas each axial anion on one side forms a bond with an equatorial Pb cation, and on the other side, it is only weakly bonded to the organic layer. Substitution at equatorial sites will therefore lead to larger structural changes than substitution at axial sites. A reduced volume leads to an enhanced chemical bonding between the halide anions with surrounding cations, which results in a lower enthalpy.
By applying the quasichemical approximation (QCA) 27−29 of binary alloying theory to the mixing enthalpies calculated at discrete x, we obtain the mixing enthalpy ΔU(x, T) as a continuous function of Br concentration x for different temperatures (lines in Figure 1a). Taking additionally the mixing entropy ΔS(x, T) into account, according to eq 2 in "Methods", yields the compositional mixing free energy ΔF(x, T) per f.u. at different temperatures, displayed in Figure 1c. As a reference, the mixing enthalpy and free energy of 3D MAPb(I 1−x Br x ) 3 are reproduced in Section S2 in the Supporting Information.
The composition−temperature, x−T, phase diagram of a mixed halide perovskite in the dark can, analogously to ordinary binary mixtures, be constructed by collecting the points of common tangent (binodal) and the inflection points (spinodal) of the compositional mixing free energy in x−T space. 27,29 In Figure 2a,b, we show the phase diagrams in the dark of 3D MAPb(I 1−x Br x ) 3 (reproduced from ref 27) and 2D PEA 2 Pb(I 1−x Br x ) 4 , respectively. The blue curve in the phase diagrams is the binodal, separating the metastable region (gray) from the stable region (white). The red curve is the spinodal, separating the unstable region (pink) from the metastable region. The position where the binodal and spinodal meet is the critical point (x c , T c ). The predicted critical temperature T c of 2D PEA 2 Pb(I 1−x Br x ) 4 is about 161 K, which is much lower than that of 3D MAPb(I 1−x Br x ) 3 (266 K). 27 This shows that PEA 2 Pb(I 1−x Br x ) 4 is thermodynamically much more stable in the dark. The superior phase stability of PEA 2 Pb(I 1−x Br x ) 4 is explained by the favorable I−Br distribution, as discussed above and shown in Figure 1. Since  in this favorable distribution the I and Br anions are already well demixed (the I anions prefer to be at axial sites and the Br anions prefer to be at equatorial sites), the enthalpic driving force for a further demixing is strongly reduced.
The presence of photocarriers under illumination requires addition of a photocarrier free energy contribution to the compositional mixing free energy. 27 Assuming segregation into two phases with different Br concentrations x 1 and x 2 , this free energy contribution is equal to the sum over the two phases of the number of photocarriers in each phase multiplied by the band gap of the phase; see eq 3 in "Methods". The band gaps of PEA 2 Pb(I 1−x Br x ) 4 and MAPb(I 1−x Br x ) 3 are taken from experiment (see Section S3 in the Supporting Information). The band gap of PEA 2 Pb(I 1−x Br x ) 4 can be well described by 21 which interpolates between the band gap of 2.37 eV for PEA 2 PbI 4 and 3.03 eV for PEA 2 PbBr 4 . The band gap of MAPb(I 1−x Br x ) 3 is smaller and described by the function E g (x) 30 with band gaps of 1.57 eV for MAPbI 3 and 2.29 eV for MAPbBr 3 . The dependence of the band gap on the Br concentration x in PEA 2 Pb(I 1−x Br x ) 4 is slightly smaller than in MAPb(I 1−x Br x ) 3 , but we will see that this small difference has an important effect on the phase diagrams under illumination.
The photocarrier densities in the two phases are obtained from a thermally governed band-gap-dependent redistribution over the two phases and a balance between the photocarrier generation and annihilation processes in the two phases; see eqs 4 and 5 in "Methods". We make the simplifying assumption that the photocarrier generation rate G is the same in the two phases and is for a thin film given by G = IαV/ hν, where I is the illumination intensity (I ≈ 100 mW cm −2 for 1 Sun), α is the absorption coefficient, V is the volume per f.u., and hν is the photon energy. The annihilation of photocarriers is characterized by monomolecular and bimolecular recombination in the two phases, where the rate constants, given by the inverse photocarrier lifetime 1/τ and k, respectively, are assumed to be phase-independent. For both MAPb(I 1−x Br x ) 3 and PEA 2 Pb(I 1−x Br x ) 4 , we take the values α = 10 −5 cm −1 , hν = 3 eV, 26  The halide segregation in mixed I−Br perovskites under illumination is a consequence of a decreased free energy by accumulation of photocarriers in a low-band-gap nucleated Irich phase, 26,27 where the driving force for halide demixing is the band gap difference between the parent mixed phase and the nucleated low-band-gap I-rich phase. 26 The band gap differences for different halide compositions in MAPb-(I 1−x Br x ) 3 are mainly caused by changes in the energy of the valence band maximum, which increases with increasing I concentration. 26 In PEA 2 Pb(I 1−x Br x ) 4 , the energy of the valence band maximum increases and the energy of the conduction band minimum decreases with increasing I concentration. 34 We thus conclude that in MAPb(I 1−x Br x ) 3 it will be mainly the photogenerated holes, while in PEA 2 Pb-(I 1−x Br x ) 4 , it will be both the photogenerated electrons and holes that can reduce their free energy by funneling into I-rich domains. In the presence of illumination, the usual method of finding binodals and spinodals�used to obtain the phase diagram in the dark of Figure 2�is not applicable, and instead, a more sophisticated procedure should be used to find the minima of the total (compositional plus photocarrier) free energy under the constraints ϕ 1 + ϕ 2 = 1 (ϕ 1 and ϕ 2 are the volume fractions of the two phases) and ϕ 1 x 1 + ϕ 2 x 2 = x. 27 Figure 3a−f shows the composition−temperature phase diagrams for PEA 2 Pb(I 1−x Br x ) 4 and MAPb(I 1−x Br x ) 3 at increasing illumination intensities I = 0.1, 1, and 10 Sun, taking τ = 100 ns for both perovskites. In comparison to the phase diagrams in the dark (see Figure 2), the spinodals in both perovskites only slightly change by the illumination. By contrast, the binodals change considerably. As already found in our previous analysis, 27 for 3D MAPb(I 1−x Br x ) 3 , two types of binodals are obtained, a compositional binodal (blue curve) and a light-induced binodal (green curve). When the compositional binodal is crossed by increasing x or decreasing T, nucleation of a phase that is more Br-rich than the parent phase becomes favorable, as indicated by the dashed blue line. When the light-induced binodal is crossed by increasing x or decreasing T, a nearly I-pure phase is nucleated, as indicated by the dashed green line. The location where the compositional and light-induced binodals meet was suggested to be a triple point where two phases with different halide compositions may be nucleated out of the parent phase, 27 as indicated by the three differently colored dots. In 2D PEA 2 Pb(I 1−x Br x ) 4 , only the light-induced binodal exists under the investigated illumination intensities for a photocarrier lifetime of 100 ns.
By comparing Figure 3d−f to Figure 3a−c, we see that at 300 K (room temperature) the light-induced binodals of PEA 2 Pb(I 1−x Br x ) 4 occur at a higher Br concentration x than those of MAPb(I 1−x Br x ) 3 . This means that PEA 2 Pb(I 1−x Br x ) 4 is thermodynamically more stable than MAPb(I 1−x Br x ) 3 for comparable illumination intensities and photocarrier lifetimes. This increased photostability is caused by the smaller band gap difference between the parent and nucleated phase in PEA 2 Pb(I 1−x Br x ) 4 as compared to MAPb(I 1−x Br x ) 3 , which leads to a smaller driving force for halide demixing. However, this effect alone cannot explain the absence of halide segregation for x = 0.5 in PEA 2 Pb(I 1−x Br x ) 4 at room temperature and under 1 Sun illumination, 17,18 since Figure  3e shows that PEA 2 Pb(I 0.5 Br 0.5 ) 4 is metastable under these conditions (gray region), so that halide demixing is still expected to occur. Figure 3g−i shows for PEA 2 Pb(I 1−x Br x ) 4 results comparable to Figure 3d−f, but for the appropriate experimentally determined photocarrier lifetime τ = 1 ns. 31,33 We observe that the decrease in lifetime from 100 to 1 ns has almost no effect on the spinodals but leads to a strong shift of the binodals to higher Br concentration x. Like in 3D MAPb-(I 1−x Br x ) 3 , a compositional binodal and a triple point appear. The increased photostability by the reduction of the photocarrier lifetime is caused by the reduced concentration of photocarriers and the concurring reduced driving force for phase segregation. The reduction in lifetime by a factor 100 roughly corresponds to a reduction of the illumination intensity by the same factor, as illustrated by the similarity of the phase diagrams of Figure 3d,i.
When the light-induced binodal is crossed, a metastable region is entered, leading to a threshold for halide demixing in x−T−I phase space, beyond which demixing is expected. 27 The threshold illumination intensity in PEA 2 Pb(I 1−x Br x ) 4 for x = 0.5, T = 300 K, and τ = 1 ns is calculated to be about I = 90 Sun, substantially higher than in MAPb(I 1−x Br x ) 3 (about 0.02 Sun). 27 The threshold Br concentration x in PEA 2 Pb(I 1−x Br x ) 4 at 300 K under 1 Sun illumination is about 0.7, which is a factor of about 2 larger than in MAPb(I 1−x Br x ) 3 . 27 Because x = 0.5 is below the threshold Br concentration for halide demixing, PEA 2 Pb(I 1−x Br x ) 4 is predicted by our theory to be thermodynamically stable at 300 K under 1 Sun illumination, in accordance with the experimental observations. 17, 18 We note that this prediction is obtained within an equilibrium thermodynamic theory and should therefore be contrasted to the kinetic explanation of increased photostability based on an increased barrier for halide migration. 17 Recently, it has been reported that the Dion-Jacobson 2D perovskite (PDMA)Pb-(I 1−x Br x ) 4 (PDMA: 1,4-phenylenedimethanammonium) demixes under illumination for x = 0.5. 35 This might be ascribed to a higher photocarrier lifetime in (PDMA)Pb(I 1−x Br x ) 4 in comparison to PEA 2 Pb(I 1−x Br x ) 4 , possibly because of a lower concentration of defects. The bivalent nature of the PDMA cation might result in a better crystallinity of (PDMA)Pb-(I 1−x Br x ) 4 as compared to PEA 2 Pb(I 1−x Br x ) 4 , where PEA is monovalent. To resolve this issue, it would be helpful if the photocarrier lifetime in (PDMA)Pb(I 1−x Br x ) 4 is determined.
In summary, using a unified thermodynamic theory, we have proposed an equilibrium thermodynamic origin of the superior phase stability of the 2D mixed halide perovskite PEA 2 Pb-(I 1−x Br x ) 4 as compared to its 3D counterpart perovskite MAPb(I 1−x Br x ) 3 . We have found that PEA 2 Pb(I 1−x Br x ) 4 is thermodynamically more stable than MAPb(I 1−x Br x ) 3 , both in the dark and under illumination. Several factors can explain this difference. The improved phase stability of PEA 2 Pb-(I 1−x Br x ) 4 in the dark can be explained by an energetically favorable I−Br distribution, where I and Br anions are preferably located on different types of lattice sites. The smaller band gap difference of the mixed parent phase and the nucleated low-band-gap phase, and particularly a much shorter photocarrier lifetime, are identified as the reasons for the enhanced phase stability of PEA 2 Pb(I 1−x Br x ) 4 under illumination. These findings provide important fundamental insight into the suppressed halide segregation in PEA 2 Pb(I 1−x Br x ) 4 . Such insight is critical in the quest for long-term stable mixed halide perovskites for use in optoelectronic devices.
inorganic Pb−I octahedral layers in the equatorial plane and two organic PEA bilayers intercalating along the axial direction. 7 We then replace I anions by Br anions at different Br concentrations x = 0, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8, 1. For each possible configuration, we take the same halide distribution for the two parallel inorganic layers. The total number of possible configurations then becomes 2 8 = 256. With a perfect D 4h symmetry for each inorganic layer, the total number of inequivalent configurations is reduced to 45. The small deviation from D 4h symmetry due to the presence of the spacer PEA cations is not important because the PEA cation is not incorporated into the inorganic layer, unlike in the 3D case. We note that for each configuration, the whole structure is optimized without symmetry restrictions.
The total energy calculations are performed within density functional theory (DFT), using the projector augmented wave (PAW) 36 method as implemented in the Vienna Ab initio Simulation Package (VASP). 37 The used electronic exchangecorrelation interaction is described by the Perdew−Burke− Ernzerhof (PBE) functional within the generalized gradient approximation (GGA). 38 We use 6 × 6 × 2 k-point Brillouin zone samplings and a plane-wave kinetic energy cutoff of 500 eV. The D3 correction 39 is used to describe the van der Waals interactions between the organic PEA bilayers and the inorganic layers. The shape, volume, and atomic positions of each possible configuration are fully optimized. The structure files of the optimized configurations can be found in the Supporting Information. Energy and force convergence criteria of 0.01 meV and 0.01 eV/Å, respectively, are used in all calculations.
Calculation of the Mixing Free Energy in the Dark. The mixing enthalpy per f.u. ΔU j of inequivalent I−Br configurations j = 1, 2, ..., J for 2D PEA 2 Pb(I 1−x Br x ) 4 is calculated by where E j , E 1 , and E J are the total energies per f.u. of the inequivalent mixed I−Br, the pure I, and the pure Br configurations, respectively. The results are displayed by circles in Figure 1a. We apply the quasichemical approximation (QCA) 28 to obtain the Helmholtz compositional free energy ΔF(x, T) as functions of the Br concentration x and temperature T, where ΔU(x, T) and ΔS(x, T) are the mixing enthalpy and entropy, respectively. Further computational details can be found in ref 27.
Calculation of the Free Energy under Illumination. Like in ref 27, we minimize the total free energy per f.u. under illumination, which consists of the sum over the two phases of a compositional and a photocarrier free energy: Here, ϕ 1 and ϕ 2 are the volume fractions of the two phases, and x 1 and x 2 are the Br concentrations of the two phases, which have different band gaps E g (x 1 ) and E g (x 2 ). Neglecting the small volume difference per f.u. between the two phases yields the conditions ϕ 1 + ϕ 2 = 1 and ϕ 1 x 1 + ϕ 2 x 2 = x.
Depending on the band gaps of the two phases, the photocarriers thermally redistribute over the two phases with different densities (numbers of photocarriers per f.u.) n 1 and n 2 . Demanding local charge neutrality, the photocarrier densities correspond in each phase to the densities of photogenerated electrons as well as holes. Since n 1 , n 2 ≪ 1, we can use Boltzmann statistics: where k B T is the thermal energy. We assume that the diffusion length of photocarriers is larger than the feature size of domains so that we can take the distribution of photocarriers in each phase to be uniform. In equilibrium, the total generation rates of photocarriers are equal to the total annihilation rates by monomolecular and bimolecular recombination in the two phases: The techniques used in finding the�local and global� minima of the total free energy equation (eq 3), needed to obtain the binodals and spinodals in Figures 2 and 3, and the thresholds for halide demixing are the same as in ref 27